The approximate half life of titanium-44 is 63 years. How long will it take to lose 
a.) 50% of its titanium-44
b.)75% of its titanium-44

The decay of a radioactive substance, such as titanium-44, follows an exponential decay model. The amount of substance remaining after a given time can be described by the formula:

\[
N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}
\]

Where:
- \( N(t) \) is the remaining quantity of the substance after time \( t \),
- \( N_0 \) is the initial quantity of the substance,
- \( T_{1/2} \) is the half-life of the substance,
- \( t \) is the time elapsed.

Given:
- Half-life \( T_{1/2} = 63 \) years.

### a) To lose 50% of titanium-44:
This corresponds to one half-life, since losing 50% means half the original amount remains.

So:
\[
t_{50\%} = T_{1/2} = 63 \text{ years}
\]

### b) To lose 75% of titanium-44:
To lose 75% means that 25% of the original amount remains. We need to solve for \( t \) when \( N(t) = 0.25N_0 \).

\[
0.25 = \left(\frac{1}{2}\right)^{\frac{t}{63}}
\]

Taking the logarithm of both sides:

\[
\log(0.25) = \frac{t}{63} \times \log\left(\frac{1}{2}\right)
\]

Solve for \( t \):

\[
t = 63 \times \frac{\log(0.25)}{\log\left(\frac{1}{2}\right)}
\]

Let's calculate \( t \) for part b.### Results:
a) It will take 63 years to lose 50% of titanium-44.

b) It will take 126 years to lose 75% of titanium-44.

Do you want more details or have any questions?

Here are five related questions for further exploration:
1. How much titanium-44 remains after 100 years?
2. What is the relationship between half-life and the time to lose 90% of titanium-44?
3. How would the time change if the half-life of titanium-44 were shorter?
4. How does the decay of titanium-44 compare to other isotopes with shorter half-lives?
5. What are some practical applications of knowing the half-life of a substance?

**Tip:** Remember that the half-life is a constant, meaning that the same amount of time will always be required to reduce the quantity by half, regardless of how much remains.

Explore how to calculate radioactive decay problems involving titanium-44. Learn the time it takes to lose 50% and 75% of titanium-44 using its half-life of 63 years. Detailed step-by-step solutions included.

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